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Bloch-like waves in random-walk potentials based on supersymmetry.

Yu S, Piao X, Hong J, Park N - Nat Commun (2015)

Bottom Line: Bloch's theorem was a major milestone that established the principle of bandgaps in crystals.Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps.Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with 'Bloch-like eigenstates'.

View Article: PubMed Central - PubMed

Affiliation: Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea.

ABSTRACT
Bloch's theorem was a major milestone that established the principle of bandgaps in crystals. Although it was once believed that bandgaps could form only under conditions of periodicity and long-range correlations for Bloch's theorem, this restriction was disproven by the discoveries of amorphous media and quasicrystals. While network and liquid models have been suggested for the interpretation of Bloch-like waves in disordered media, these approaches based on searching for random networks with bandgaps have failed in the deterministic creation of bandgaps. Here we reveal a deterministic pathway to bandgaps in random-walk potentials by applying the notion of supersymmetry to the wave equation. Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps. Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with 'Bloch-like eigenstates'.

No MeSH data available.


Related in: MedlinePlus

2D SUSY-transformed potentials with bandgaps maintained.The evolutions of the potential profiles following the application of SUSY transformations to the x and y axes are shown: (a) original and (b) x axis SUSY transformed (5th). (c) The y axis SUSY-transformed (5th), and (d) x and y symmetric SUSY-transformed (x axis: 5th; y axis: 5th) potentials. The spatial profiles of the potentials for 0° (x axis), 45° and 90° (y axis) are also overlaid in a–d. (e) The eigenvalues of the SUSY-transformed potentials as a function of the modal numbers. The grey regions denote bandgaps. The total SUSY number is the sum of the number of SUSY transformations for x and y axes (that is, 5 for both b and c and 10 for d). (f,g) The Hurst exponents for different directions of the 2D potentials that are (f) highly anisotropic (x axis: 5th, y axis: 0th) and (g) quasi-isotropic (x axis: 5th; y axis: 5th) SUSY transformations. The grey symbols in f and g are the Hurst exponents of the original potential.
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f5: 2D SUSY-transformed potentials with bandgaps maintained.The evolutions of the potential profiles following the application of SUSY transformations to the x and y axes are shown: (a) original and (b) x axis SUSY transformed (5th). (c) The y axis SUSY-transformed (5th), and (d) x and y symmetric SUSY-transformed (x axis: 5th; y axis: 5th) potentials. The spatial profiles of the potentials for 0° (x axis), 45° and 90° (y axis) are also overlaid in a–d. (e) The eigenvalues of the SUSY-transformed potentials as a function of the modal numbers. The grey regions denote bandgaps. The total SUSY number is the sum of the number of SUSY transformations for x and y axes (that is, 5 for both b and c and 10 for d). (f,g) The Hurst exponents for different directions of the 2D potentials that are (f) highly anisotropic (x axis: 5th, y axis: 0th) and (g) quasi-isotropic (x axis: 5th; y axis: 5th) SUSY transformations. The grey symbols in f and g are the Hurst exponents of the original potential.

Mentions: The derivation in the Methods section (Equations 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21) starting from equation (5) demonstrates that the procedure of the 1D SUSY transformation can be applied to a 2D potential for each x and y axes independently, when the potential satisfies the condition of Vo(x,y)=Vox(x)+Voy(y). We also note that serial 2D SUSY transformations are possible because the form of Vo(x,y)=Vox(x)+Voy(y) is preserved during the transformation, consequently deriving a family of 2D quasi-isospectral potentials. Figure 5 shows an example of SUSY transformations in 2D potentials, maintaining Bloch-like eigenstates. Both the x and y axes cross-sections of the 2D original potential Vo(x,y)=Vox(x)+Voy(y) have profiles of N=8 binary sequences (Fig. 5a), as defined in Fig. 1. Following the procedure of equations (17, 18, 19, 20, 21) in the Methods section, we apply SUSY transformations to the x and y axes separately, achieving the highly anisotropic shape of the potential as shown in Fig. 5b (the 5th x axis SUSY-transformed potential) and Fig. 5c (the 5th y axis SUSY-transformed potential). It is evident that this anisotropy can be controlled by changing the number of SUSY transformations for the x and y axes independently, and the isotropic application of SUSY transformations recovers the isotropic potential shape (Fig. 5d). Regardless of the number of SUSY transformations and their anisotropic implementations, the region of bandgaps of the original potential is always preserved (Fig. 5e). Interestingly, the annihilation by 2D SUSY transformation occurs not only in the ground state but also in all of the excited states sharing a common 1D ground-state profile (for details see the Methods section, Supplementary Note 3 and Supplementary Fig. 9). Consequently, the width of the bandgap can be slightly changed owing to the annihilation of some excited states near the bandgap.


Bloch-like waves in random-walk potentials based on supersymmetry.

Yu S, Piao X, Hong J, Park N - Nat Commun (2015)

2D SUSY-transformed potentials with bandgaps maintained.The evolutions of the potential profiles following the application of SUSY transformations to the x and y axes are shown: (a) original and (b) x axis SUSY transformed (5th). (c) The y axis SUSY-transformed (5th), and (d) x and y symmetric SUSY-transformed (x axis: 5th; y axis: 5th) potentials. The spatial profiles of the potentials for 0° (x axis), 45° and 90° (y axis) are also overlaid in a–d. (e) The eigenvalues of the SUSY-transformed potentials as a function of the modal numbers. The grey regions denote bandgaps. The total SUSY number is the sum of the number of SUSY transformations for x and y axes (that is, 5 for both b and c and 10 for d). (f,g) The Hurst exponents for different directions of the 2D potentials that are (f) highly anisotropic (x axis: 5th, y axis: 0th) and (g) quasi-isotropic (x axis: 5th; y axis: 5th) SUSY transformations. The grey symbols in f and g are the Hurst exponents of the original potential.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4595658&req=5

f5: 2D SUSY-transformed potentials with bandgaps maintained.The evolutions of the potential profiles following the application of SUSY transformations to the x and y axes are shown: (a) original and (b) x axis SUSY transformed (5th). (c) The y axis SUSY-transformed (5th), and (d) x and y symmetric SUSY-transformed (x axis: 5th; y axis: 5th) potentials. The spatial profiles of the potentials for 0° (x axis), 45° and 90° (y axis) are also overlaid in a–d. (e) The eigenvalues of the SUSY-transformed potentials as a function of the modal numbers. The grey regions denote bandgaps. The total SUSY number is the sum of the number of SUSY transformations for x and y axes (that is, 5 for both b and c and 10 for d). (f,g) The Hurst exponents for different directions of the 2D potentials that are (f) highly anisotropic (x axis: 5th, y axis: 0th) and (g) quasi-isotropic (x axis: 5th; y axis: 5th) SUSY transformations. The grey symbols in f and g are the Hurst exponents of the original potential.
Mentions: The derivation in the Methods section (Equations 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21) starting from equation (5) demonstrates that the procedure of the 1D SUSY transformation can be applied to a 2D potential for each x and y axes independently, when the potential satisfies the condition of Vo(x,y)=Vox(x)+Voy(y). We also note that serial 2D SUSY transformations are possible because the form of Vo(x,y)=Vox(x)+Voy(y) is preserved during the transformation, consequently deriving a family of 2D quasi-isospectral potentials. Figure 5 shows an example of SUSY transformations in 2D potentials, maintaining Bloch-like eigenstates. Both the x and y axes cross-sections of the 2D original potential Vo(x,y)=Vox(x)+Voy(y) have profiles of N=8 binary sequences (Fig. 5a), as defined in Fig. 1. Following the procedure of equations (17, 18, 19, 20, 21) in the Methods section, we apply SUSY transformations to the x and y axes separately, achieving the highly anisotropic shape of the potential as shown in Fig. 5b (the 5th x axis SUSY-transformed potential) and Fig. 5c (the 5th y axis SUSY-transformed potential). It is evident that this anisotropy can be controlled by changing the number of SUSY transformations for the x and y axes independently, and the isotropic application of SUSY transformations recovers the isotropic potential shape (Fig. 5d). Regardless of the number of SUSY transformations and their anisotropic implementations, the region of bandgaps of the original potential is always preserved (Fig. 5e). Interestingly, the annihilation by 2D SUSY transformation occurs not only in the ground state but also in all of the excited states sharing a common 1D ground-state profile (for details see the Methods section, Supplementary Note 3 and Supplementary Fig. 9). Consequently, the width of the bandgap can be slightly changed owing to the annihilation of some excited states near the bandgap.

Bottom Line: Bloch's theorem was a major milestone that established the principle of bandgaps in crystals.Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps.Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with 'Bloch-like eigenstates'.

View Article: PubMed Central - PubMed

Affiliation: Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea.

ABSTRACT
Bloch's theorem was a major milestone that established the principle of bandgaps in crystals. Although it was once believed that bandgaps could form only under conditions of periodicity and long-range correlations for Bloch's theorem, this restriction was disproven by the discoveries of amorphous media and quasicrystals. While network and liquid models have been suggested for the interpretation of Bloch-like waves in disordered media, these approaches based on searching for random networks with bandgaps have failed in the deterministic creation of bandgaps. Here we reveal a deterministic pathway to bandgaps in random-walk potentials by applying the notion of supersymmetry to the wave equation. Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps. Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with 'Bloch-like eigenstates'.

No MeSH data available.


Related in: MedlinePlus